Wedge product of forms

Definition

Let $U \subseteq \mathbb{R}^n$ be open. For indices $j_1, \dots, j_m \in \{1, \dots, n\}$, let $dx_{j_1} \wedge \dots \wedge dx_{j_m}$ be $0$ if two indices are equal. Otherwise it is $\operatorname{sgn}(\sigma) \, dx_I$, where $I$ is the increasing rearrangement of the indices and $\sigma$ is the permutation that sorts them. For differential forms $\alpha = \sum_I f_I \, dx_I$ of degree $k$ and $\beta = \sum_J g_J \, dx_J$ of degree $l$ on $U$, the wedge product is the $(k + l)$-form $$\alpha \wedge \beta = \sum_{I, J} f_I \, g_J \; dx_{i_1} \wedge \dots \wedge dx_{i_k} \wedge dx_{j_1} \wedge \dots \wedge dx_{j_l}.$$

Remarks

The product is bilinear and associative. It is graded commutative: $\beta \wedge \alpha = (-1)^{k l} \, \alpha \wedge \beta$. In particular, $dx_i \wedge dx_j = -dx_j \wedge dx_i$ and $dx_i \wedge dx_i = 0$. For a $0$-form $f$, the product $f \wedge \beta$ is the form $f \beta$. In the plane, $(a \, dx + b \, dy) \wedge (c \, dx + d \, dy) = (a d - b c) \, dx \wedge dy$. The coefficient is the determinant, which is the signed area of the parallelogram spanned by $(a, b)$ and $(c, d)$. For 1-forms $\alpha$ and $\beta$ and vectors $v, w$, $(\alpha \wedge \beta)(v, w) = \alpha(v) \beta(w) - \alpha(w) \beta(v)$.

Explanations

L1 human

The wedge product is a multiplication of forms in which order matters up to sign, and a form times itself is zero. It is the algebra behind determinants, signed areas and signed volumes.